97,414 research outputs found
Strong Cosmic Censorship with Bounded Curvature
In this paper we propose a weaker version of Penrose's much heeded Strong
Cosmic Censorship (SCC) conjecture, asserting inextentability of maximal Cauchy
developments by manifolds with Lipschitz continuous Lorentzian metrics and
Riemann curvature bounded in . Lipschitz continuity is the threshold
regularity for causal structures, and curvature bounds rule out infinite tidal
accelerations, arguing for physical significance of this weaker SCC conjecture.
The main result of this paper, under the assumption that no extensions exist
with higher connection regularity , proves in the
affirmative this SCC conjecture with bounded curvature for sufficiently
large, ( with uniform bounds, without uniform bounds)
Weighted bounds for multilinear operators with non-smooth kernels
Let be a multilinear integral operator which is bounded on certain
products of Lebesgue spaces on . We assume that its associated
kernel satisfies some mild regularity condition which is weaker than the usual
H\"older continuity of those in the class of multilinear Calder\'on-Zygmund
singular integral operators. In this paper, given a suitable multiple weight
, we obtain the bound for the weighted norm of multilinear operators
in terms of . As applications, we exploit this result to obtain
the weighted bounds for certain singular integral operators such as linear and
multilinear Fourier multipliers and the Riesz transforms associated to
Schr\"odinger operators on . Our results are new even in the
linear case
Solving linear parabolic rough partial differential equations
We study linear rough partial differential equations in the setting of [Friz
and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear
parabolic partial differential equation driven by a deterministic rough path
of H\"older regularity with . Based on a stochastic representation of the solution of the rough
partial differential equation, we propose a regression Monte Carlo algorithm
for spatio-temporal approximation of the solution. We provide a full
convergence analysis of the proposed approximation method which essentially
relies on the new bounds for the higher order derivatives of the solution in
space. Finally, a comprehensive simulation study showing the applicability of
the proposed algorithm is presented
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